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1. Geodesic planes in geometrically finite acylindrical -manifolds

   https://doi.org/10.1017/etds.2021.19  

   BENOIST, YVES; OH, HEE (February 2022, Ergodic Theory and Dynamical Systems)

   Abstract Let M be a geometrically finite acylindrical hyperbolic $$3$$ -manifold and let $M^*$ denote the interior of the convex core of M . We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J. , to appear, Preprint , 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $$3$$ -manifold $$M_0$$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $$M_0$$ . We construct a counterexample of this phenomenon when $$M_0$$ is non-arithmetic. 

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2. Geodesic Growth of Numbered Graph Products

   https://doi.org/10.46298/jgcc.2023.14.2.10019  

   Marjanski, Lindsay; Solon, Vincent; Zheng, Frank; Zopff, Kathleen (February 2023, journal of Groups, complexity, cryptology)

   In this paper, we study geodesic growth of numbered graph products; these area generalization of right-angled Coxeter groups, defined as graph products offinite cyclic groups. We first define a graph-theoretic condition calledlink-regularity, as well as a natural equivalence amongst link-regular numberedgraphs, and show that numbered graph products associated to link-regularnumbered graphs must have the same geodesic growth series. Next, we derive aformula for the geodesic growth of right-angled Coxeter groups associated tolink-regular graphs. Finally, we find a system of equations that can be used tosolve for the geodesic growth of numbered graph products corresponding tolink-regular numbered graphs that contain no triangles and have constant vertexnumbering. 

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3. Endperiodic automorphisms of surfaces and foliations

   https://doi.org/10.1017/etds.2019.56  

   CANTWELL, JOHN; CONLON, LAWRENCE; FENLEY, SERGIO R. (January 2021, Ergodic Theory and Dynamical Systems)

   We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations $${\mathcal{F}}$$ and $${\mathcal{F}}^{\prime }$$ are transverse to a common one-dimensional foliation $${\mathcal{L}}$$ whose monodromy on the non-compact leaves of $${\mathcal{F}}$$ exhibits the nice dynamics of Handel–Miller theory, then $${\mathcal{L}}$$ also induces monodromy on the non-compact leaves of $${\mathcal{F}}^{\prime }$$ exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends. 

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4. On the Reconstruction of Geodesic Subspaces of ℝ^N

   https://doi.org/10.1142/S0218195922500066  

   Fasy, Brittany Terese; Komendarczyk, Rafal; Majhi, Sushovan; Wenk, Carola (January 2022, International Journal of Computational Geometry & Applications)

   We consider the topological and geometric reconstruction of a geodesic subspace of [Formula: see text] both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for the reconstruction problem. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. This technique provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and [Formula: see text]–reach, and performs better under certain types of perturbations of the geodesic subspace. For geodesic subspaces of [Formula: see text], we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown underlying space. 

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5. Variation of holonomy for projective structures and an application to drilling hyperbolic 3-manifolds

   https://doi.org/10.1007/s10711-024-00908-0  

   Bridgeman, Martin; Bromberg, Kenneth (June 2024, Geometriae Dedicata)

   We bound the derivative of complex length of a geodesic under variation of the projective structure on a closed surface in terms of the norm of the Schwarzian in a neighborhood of the geodesic. One application is to cone-manifold deformations of acylindrical hyperbolic 3-manifolds. 

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